Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C13×#41 ⋊ C2 |
| Sylow | C13 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D13 |
| commutator, socle, cosocle | C2×C13 | C24 |
| Fitting | C13×#41 | C2 |
| hypercenter | #41 | D13 |
| hypocenter | C13 | 2−1+4 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ #41 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,18,19,17)(15,21,22,26)(16,23,24,25)(20,28,29,27);(14,17,19,18)(15,21,22,26)(16,23,24,25)(20,27,29,28);(14,16,19,24)(15,20,22,29)(17,25,18,23)(21,28,26,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,20)(17,21)(18,26)(19,22)(23,27)(24,29)(25,28)Polycyclic presentation:
2^5,13|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^12,c^b=ce77 conjugacy classes — view the full character table.
| maximal subgroup of | #11718487, #11718488, #11718862, … |
| maximal normal subgroup of | #11718487, #11718488, #11718862, … |
| maximal quotient of | #11719221, #11719223, #11719224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 224);